23 11
Article 08.2.2
Journal of Integer Sequences, Vol. 11 (2008), 2
3 6 1 47
A Note on Krawtchouk Polynomials
and Riordan Arrays
Paul Barry
School of Science
Waterford Institute of Technology
Ireland
pbarry@wit.ie
Abstract
We study links between Krawtchouk polynomials and Riordan arrays of both the ordinary kind and the exponential kind. We derive summation formulas for values of the Krawtchouk polynomials using theA-sequences of the Riordan arrays.
1
Introduction
The Krawtchouk polynomials play an important role in various areas of mathematics. No-table applications occur in coding theory [11], association schemes [4], and in the theory of group representations [21,22].
In this note, we explore links between the Krawtchouk polynomials and Riordan arrays, of both ordinary and exponential type, and we study integer sequences defined by evaluating the Krawtchouk polynomials at different values of their parameters.
The link between Krawtchouk polynomials and exponential Riordan arrays is implicitly contained in the umbral calculus approach to certain families of orthogonal polynomials. We shall look at these links explicitly in the following.
2
Riordan arrays
The Riordan group [12, 17, 19], is a set of infinite lower-triangular integer matrices, where each matrix is defined by a pair of generating functions g(x) = g0 +g1x+g2x2+. . . and
f(x) = f1x+f2x2+. . . where f1 6= 0. We sometimes write f(x) =xh(x) where h(0) 6= 0.
The associated matrix is the matrix whose i-th column is generated by g(x)f(x)i (the first
column being indexed by 0). The matrix corresponding to the pair f, g is denoted by (g, f) orR(g, f), and is often called theRiordan array defined byg andf. Wheng0 = 1, the array
is called a monic Riordan array. The group law is given by
(g, f)∗(u, v) = (g(u◦f), v◦f). (1)
The identity for this law is I = (1, x) and the inverse of (g, f) is (g, f)−1 = (1/(g◦f¯),f¯)
where ¯f is the compositional inverse of f.
To each Riordan array as defined above is associated an integer sequence A={ai} with
a0 6= 0 such that every element dn+1,k+1 of the array (not lying in column 0 or row 0) can
be expressed as a linear combination with coefficients in A of the elements in the preceding row, starting from the preceding column:
dn+1,k+1 =a0dn,k+a1dn,k+1+a2dn,k+2+· · ·
A={ai}is called theA-sequence of the array, and its generating function may be calculated
according to
A(x) = [h(t)|t=xh(t)−1].
A Riordan array of the form (g(x), x), whereg(x) is the generating function of the sequence
un, is called the Appell array (or sometimes the sequence array) of the sequence un. Its
general term is un−k.
IfMis the matrix (g, f), andu = (u0, u1, . . .)′ is an integer sequence with ordinary generating
function U (x), then the sequence Mu has ordinary generating function g(x)U(f(x)).
Example 1. The binomial matrix B is the element ( 1
1−x, x
1−x) of the Riordan group. It
has general element nk
. For this matrix we have A(x) = 1 +x, which translates the usual defining relationship for Pascal’s triangle
n+ 1
k+ 1
=
n k
+
n k+ 1
.
More generally, Bm is the element ( 1 1−mx,
x
1−mx) of the Riordan group, with general term n
k
mn−k. It is easy to show that the inverseB−m of Bm is given by ( 1 1+mx,
x
1+mx).
Example 2. We let c(x) = 1−√1−4x
2x be the generating function of the Catalan numbers
Cn = n+11 2nn
A000108. The array (1, xc(x)) is the inverse of the array (1, x(1−x)) while
the array (1, xc(x2)) is the inverse of the array (1, x
Example 3. The row sums of the matrix (g, f) have generating functiong(x)/(1−f(x)) while the diagonal sums of (g, f) have generating function g(x)/(1−xf(x)). The row sums of the array (1, xc(x)), or A106566, are the Catalan numbers Cn since 1−xc1(x) = c(x). The
diagonal sums have g.f. 1
1−x2c(x), A132364.
The exponential Riordan group [3,6,7], is a set of infinite lower-triangular integer matri-ces, where each matrix is defined by a pair of generating functionsg(x) =g0+g1x+g2x2+. . .
and f(x) =f1x+f2x2+. . . where f1 6= 0. The associated matrix is the matrix whose i-th
column has exponential generating functiong(x)f(x)i/i! (the first column being indexed by
0). The matrix corresponding to the pairf, gis denoted by [g, f]. It ismonic ifg0 = 1. The
group law is then given by
[g, f]∗[h, l] = [g(h◦f), l◦f].
The identity for this law isI = [1, x] and the inverse of [g, f] is [g, f]−1 = [1/(g◦f¯),f¯] where
¯
f is the compositional inverse of f.
IfMis the matrix [g, f], andu={un}is an integer sequence with exponential generating
function U (x), then the sequence Mu has exponential generating function g(x)U(f(x)). Thus the row sums of the array [g, f] are given byg(x)ef(x) since the sequence 1,1,1, . . .has
exponential generating function ex.
As an element of the group of exponential Riordan arrays, we have B = [ex, x]. By
the above, the exponential generating function of its row sums is given by exex = e2x, as
expected.
Riordan group techniques have been used to provide alternative proofs of many binomial identities that originally appeared in works such as [13,14]. See, for instance, [20, 19].
3
Krawtchouk polynomials
We follow [15] in defining the Krawtchouk polynomials. They form an important family of orthogonal polynomials [5,16,23]. Thus the Krawtchouk polynomials will be considered to be the special caseβ =−N,c= p−p1,p+q= 1 of the Meixner polynomials of the first kind, which form the Sheffer sequence for
g(t) =
1−c
1−cet
β
,
f(t) = 1−e
t
c−1−et.
Essentially, this says that the Meixner polynomials of the first kind are obtained by operating on the vector (1, x, x2, x3, . . .)′ by the exponential Riordan array [g(t), f(t)]−1, since
[g, f]−1 =
1
g◦f¯,f¯
and
1
g◦f¯,f¯
ext= 1
g◦f¯e
which is the defining expression for the Sheffer sequence associated tog and f. In order to work with this expression, we calculate [g, f]−1 as follows. Firstly,
¯
Then we have
g( ¯f(t)) =
Thus we arrive at
[g, f]−1 =
Extracting the coefficient of tk in this expression, we obtain
Scaling by pk, we thus obtain
We use the notation
κ(p)
for the Krawtchouk polynomial with parametersN and p. This can be expressed in hyper-geometric form as
κ(np)(x, N) = (−1)
The form of [g, f]−1 allows us to make some interesting deductions. For instance, if we write
[g(t), f(t)]−1 =
is the Stirling array of the first kind.
The matrix P[p]−N ∗Lah[p−1]∗s = h(1−pt)N,log1−(p−1)t
1−pt
i
is of course a monic exponential Riordan array. If its general term is T(n, k), then that of the corresponding array [g, f]−1 is given by T(n, k)/pn.
The above matrix factorization indicates that the Krawtchouk polynomials can be ex-pressed as combinations of the Stirling polynomials of the first kind 1, x, x(x+ 1), x(x2 +
3x+ 2), x(x3 + 6x2+ 11x+ 6), . . ..
Example 4. Taking N = −1 and p = 2 we exhibit an interesting property of the
matrix h(1−pt)N,log1−(p−1)t easy calculation shows that
We recall that the Binomial matrix with general term n
are similar, with 1 1−2t,log
1−t
1−2t
serving as matrix of change of basis for the similarity.
4
Krawtchouk polynomials and Riordan arrays
In this section, we shall use the following notation, where we define a variant on the poly-nomial family κ(np)(x, N). Thus we let
We then have
K(n, k, x, q) = [tk](1−t)x(1 + (q−1)t)n−x
,
which implies that
K(N, k, N −x, q) = [tk](1
We shall see in the sequel that by varying the parametersn, k, xandq, we can obtain families of (ordinary) Riordan arrays defined by the corresponding Krawtchouk expressions.
Example 5. We first look at the term K(k, n−k, r, q). We have
But this last term is the general term of the Riordan array
Riordan array, which is given by
The A-sequence of the array (2) is given by
A(x) = 1 +
p
1 + 4(q−1)x
2 .
Thus
a0 = 1, an= (−1)n−1(q−1)nCn−1.
With these values, we therefore have
K(k+1, n−k, r, q) =K(k, n−k, r, q)+a1K(k+1, n−k−1, r, q)+a2K(k+2, n−k−2, r, q)+. . .
Example 6. We next look at the family defined by (−1)kK(n, k, k, q). We have
(−1)kK(n, k, k, q) = (
−1)k k
X
j=0
(−1)j
k j
n−k k−j
(q−1)k−j
=
k
X
j=0
k j
n−k k−j
(1−q)k−j
=
n−k
X
j=0
k j
n−k j
(1−q)j.
Using the results of [1], we see that these represent the family of Riordan arrays
1 1−x,
x(1−qx) 1−x
.
The A-sequence for this array is given by
A(x) = 1 +x+
p
1 + 2x(1−2q) +x2
2 .
For example, the matrix with general term T(n, k) = (−1)kK(n, k, k,−3) is the Riordan
array 1−1x,x(1+31−xx), A081578 or
1 0 0 0 0 0 . . .
1 1 0 0 0 0 . . .
1 5 1 0 0 0 . . .
1 9 9 1 0 0 . . .
1 13 33 13 1 0 . . .
1 17 72 73 7 1 . . .
... ... ... ... ... ... ...
The A-sequence for this array has g.f. 1+x+√1+14x+x2
2 which expands to
Thus
The rows of this matrixA098593are the anti-diagonals (and a signed version of the diagonals) of the so-called Krawtchouk matrices [8,9] which are defined as the family of (N+1)×(N+1) matrices with general term
Kij(N)=
−1) is the well-known Delannoy number triangle 1
1−x, x(1+x)
1−x
A008288 given by
We have
. Taking theq-th inverse binomial transform of this array, we obtain
Reversing this equality gives us
have generating function
1
This is thus the generating function of the sum
n
, and will thus have the same row sums and central coefficients. The A-sequence of this array is simply 1 +qx, which implies that
Example 9. We now consider the expression (−1)n−kK(n−k, n−k, k, q). We have
hence these provide alternative expressions for (−1)n−kK(n
−k, n−k, k, q).
Similarly, we find
n
These matrices have the interesting property that T(2n, n;q) = 1. This is so since
Thus we have
K(n, n, n, q) = (−1)n.
The A-sequence for these arrays has generating function
A(x) = 1 + √
1 + 4qx
2
and thus we have
a0 = 1, an = (−1)n−1qnCn−1, n >0.
With these values we therefore have
(−1)n−kK(n
−k, n−k, k+ 1, q) = (−1)n−kK(n
−k, n−k, k, q)
+a1(−1)n−k−1K(n−k−1, n−k−1, k+ 1, q) +. . .
Example 10. We next look at the expression (−1)n−kK(n, n
−k, k, q). We have
(−1)n−kK(n, n
−k, k, q) = (−1)n−k n−k
X
j=0
(−1)j
k j
n−k n−k−j
(q−1)n−k−j
=
n−k
X
j=0
k j
n−k n−k−j
(1−q)n−k−j.
This is the general termT(n, k;q) of the Riordan array
1 1 + (q−1)x,
x(1 +qx) 1 + (q−1)x
.
Expressing T(n, k;q) differently allows us to write
n−k
X
j=0
k j
n−k n−k−j
(1−q)n−k−j = k
X
j=0
n j
n−j n−k−j
qj(1
−q)n−k−j.
The central coefficients of these arrays,T(2n, n;q), have generating functione(2−q)xI
0(2√1−qx)
and represent the n-th terms in the expansion of (1 + (2−q)x+ (1−q)x2)n.
The A-sequence for this family of arrays has generating function
A(x) = 1 + (1−q)x+
p
1 + 2x(1 +q) + (q−1)2x2
2 .
Expanding this asa0, a1, a2, . . . we thus obtain
(−1)n−k
Example 11. The expression K(n, n−k, N, q) is the general term of the Riordan array
This implies that
n−k
Example 12. In this example, we indicate that summing over one of the parameters can
still lead to a Riordan array. Thus the expression
n−k
X
i=0
(−1)iK(n
−k, i, n, q)
is equivalent to the general term of the Riordan array
while the expression
n−k
X
i=0
K(n−k, i, n, q)
is equivalent to the general term of the Riordan array
The A-sequence for this example is given by 1 +qx, and so for example we have
Example 13. The Riordan arrays encountered so far have all been of an elementary nature. The next example indicates that this is not always so. We make the simple change of 2nfornin the third parameter in the previous example. We then find thatPn−k
i=0(−1)
iK(n
−
k, i,2n, q) is the general term of the Riordan array
1−2x−q(2−q)x2
1 +qx ,
x
(1 +qx)2 −1
.
For instance, Pn−k
i=0(−1)iK(n−k, i,2n,1) represents the general term of the Riordan array
1 2
1 1−4x +
1 √
1−4x
,1−2x−
√ 1−4x
2x
while Pn−k
i=0(−1)iK(n−k, i,2n,2) represents the general term of
1 √
1−8x,
1−4x−√1−8x
2x
.
The A-sequence for the first array above is (1 +x)2, so that we obtain
n−k
X
i=0
(−1)iK(n
−k, i,2(n+ 1),1) =
n−k
X
i=0
(−1)iK(n
−k, i,2n,1)
+2
n−k−1 X
i=0
(−1)iK(n
−k−1, i,2n,1)
+
n−k−2 X
i=0
(−1)iK(n
−k−2, i,2n,1)
while that of the second array is (1 + 2x)2 and so
n−k
X
i=0
(−1)iK(n
−k, i,2(n+ 1),2) =
n−k
X
i=0
(−1)iK(n
−k, i,2n,2)
+4
n−k−1 X
i=0
(−1)iK(n
−k−1, i,2n,2)
+4
n−k−2
X
i=0
(−1)iK(n
−k−2, i,2n,2).
Table 1. Summary of Riordan arrays
Krawtchouk expression Riordan array g.f. for A-sequence
K(k, n−k, r, q) 1−x
1+(q−1)x
r
, x(1 + (q−1)x) 1+ √
1+4(q−1)x
2
(−1)n−kK(k, n−k, r, q) 1+x
1−(q−1)x
r
, x(1−(q−1)x) 1+ √
1−4(q−1)x
2
(−1)kK(n, k, k, q) 1 1−x,
x(1−qx) 1−x
1+x+√1+2x(1−2q)+x2
2
(−1)n−kK(n−k, n−k, k, q) 1 1−x,
x
1−qx
1 +qx
(−1)n−kK(n−k, n−k, k, q) 1
1+(q−1)x, x(1 +qx)
1+√1+4x
2
(−1)n−kK(n, n
−k, k, q) 1+(q1−1)x,1+(x(1+q−qx1))x 1+(1−q)x+
√
1+2x(1+q)+(q−1)2x2
2
K(n, n−k, N, q) (1−qx)N
1−(q−1)x, x
1−(q−1)x
1 + (q−1)x
Pn−k
i=0(−1)iK(n−k, i, n, q)
1 1−2x,
x
1−qx
1 +qx
Pn−k
i=0 K(n−k, i, n, q)
1, x
1+qx
1−qx
Pn−k
i=0(−1)iK(n−k, i,2n, q)
1−2x−q(2−q)x2
1+qx , x
(1+qx)2
−1
(1 +qx)2
5
Acknowledgements
The author would like to express his appreciation to an anonymous reviewer, whose careful reading of the manuscript has led to significant clarifications.
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2000 Mathematics Subject Classification: Primary 33C45; Secondary 11B83,11C20.
Keywords: Krawtchouk polynomials, orthogonal polynomials, Riordan arrays, integer se-quences.
(Concerned with sequencesA000108,A001850,A008288,A024495,A081578,A098593,A106566,
A132364.)
Received December 5 2007; revised version received May 8 2008. Published in Journal of Integer Sequences, June 3 2008.
